Goal:
    A : Type  P : A → Type  l : list A
  ============================
   ((∀ x : A, In x l → P x) → All P l)


DEBUG:Going to execute:
induction l

Goals:
    A : Type  P : A → Type
  ============================
   ((∀ x : A, In x [] → P x) → All P [])

  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  ============================
   ((∀ x : A, In x (a :: l) → P x) → All P (a :: l))


DEBUG:Going to execute:
cbn
DEBUG LTAC Evaluated term: l

Goals:
    A : Type  P : A → Type
  ============================
   ((∀ x : A, False → P x) → All P [])

  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  ============================
   ((∀ x : A, a = x ∨ In x l → P x) → All P (a :: l))


DEBUG:Going to execute:
constructor

Goals:
    A : Type  P : A → Type
  ============================
   ((∀ x : A, False → P x) → All P [])

  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  ============================
   ((∀ x : A, a = x ∨ In x l → P x) → All P (a :: l))


DEBUG:Going to execute:
<coq-core.plugins.ltac::constructor@0>

Goals:
  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  X : ∀ x : A, a = x ∨ In x l → P x
  ============================
   (P a)

  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  X : ∀ x : A, a = x ∨ In x l → P x
  ============================
   (All P l)


DEBUG:Going to execute:
auto

Goals:
  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  X : ∀ x : A, a = x ∨ In x l → P x
  ============================
   (P a)

  
  A : Type
  P : A → Type
  a : A
  l : list A
  IHl : (∀ x : A, In x l → P x) → All P l
  X : ∀ x : A, a = x ∨ In x l → P x
  ============================
   (All P l)


DEBUG:Going to execute:
<coq-core.plugins.ltac::auto@0> $1 $2 $3

Goals:
    A : Type  P : A → Type  x : A  e : nth_error [] 0 = Some x
  ============================
   (P x)

    A : Type  P : A → Type  i : nat  x : A  e : nth_error [] (S i) = Some x
  ============================
   (P x)


DEBUG:Going to execute:
<coq-core.plugins.ltac::discriminate@0>
DEBUG LTAC Evaluated term: h
DEBUG LTAC Evaluated term: i